Volume 2009, Article ID 494696,14pages doi:10.1155/2009/494696
Research Article
Secrecy Capacity of a Class of Orthogonal Relay
Eavesdropper Channels
Vaneet Aggarwal, Lalitha Sankar, A. Robert Calderbank (EURASIP Member),
and H. Vincent Poor
Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA
Correspondence should be addressed to Vaneet Aggarwal,[email protected] Received 1 December 2008; Accepted 9 June 2009
Recommended by Shlomo Shamai (Shitz)
The secrecy capacity of relay channels with orthogonal components is studied in the presence of an additional passive eavesdropper node. The relay and destination receive signals from the source on two orthogonal channels such that the destination also receives transmissions from the relay on its channel. The eavesdropper can overhear either one or both of the orthogonal channels. Inner and outer bounds on the secrecy capacity are developed for both the discrete memoryless and the Gaussian channel models. For the discrete memoryless case, the secrecy capacity is shown to be achieved by apartial decode-and-forward(PDF) scheme when the eavesdropper can overhear only one of the two orthogonal channels. Two new outer bounds are presented for the Gaussian model using recent capacity results for a Gaussian multiantenna point-to-point channel with a multiantenna eavesdropper. The outer bounds are shown to be tight for two subclasses of channels. The first subclass is one in which the source and relay are clustered, and the eavesdropper receives signals only on the channel from the source and the relay to the destination, for which the PDF strategy is optimal. The second is a subclass in which the source does not transmit to the relay, for which a noise-forwarding strategy is optimal.
Copyright © 2009 Vaneet Aggarwal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
In wireless networks for which nodes can benefit from cooperation and packet-forwarding, there is also a need to preserve the confidentiality of transmitted information from untrusted nodes. Information privacy in wireless networks has traditionally been the domain of the higher layers of the protocol stack via the use of cryptographically secure schemes. In his seminal paper on the three-node wiretap channel, Wyner showed that perfect secrecy of transmitted data from the source node can be achieved when the physical channel to the eavesdropper is noisier than the channel to the intended destination, that is, when the channel is a degraded broadcast channel [1]. This work was later extended by Csisz´ar and K¨orner to all broadcast channels with confidential messages, in which the source node sends common information to both the destination and the wiretapper and confidential information only to the destination [2].
In this paper, we study the secrecy capacity of a relay channel with orthogonal components in the presence of a passive eavesdropper node. The orthogonality comes from the fact that the relay and destination receive signals from the source on orthogonal channels; furthermore, the destination also receives transmissions from the relay on its (the desti-nation’s) channel. The orthogonal model implicitly imposes a half-duplex transmission and reception constraint on the relay. For this channel, in the absence of an eavesdropper, El Gamal and Zahedi showed that apartial decode-and-forward
(PDF) strategy, in which the source transmits two messages on the two orthogonal channels and the relay decodes its received signal, achieves the capacity [14].
We study the secrecy capacity of this channel for both the discrete memoryless and Gaussian channel models. As a first step toward this, we develop a PDF strategy for the full-duplex relay eavesdropper channel and extend it to the orthogonal model. Further, since the eavesdropper can receive signals from either orthogonal channel or both, three cases arise in the development of the secrecy capacity. We specialize the outer bounds developed in [11] for the orthogonal case and show that, for the discrete memoryless channel, PDF achieves the secrecy capacity for the two cases where the eavesdropper receives signals in only one of the two orthogonal channels.
For the Gaussian model, we develop two new outer bounds using recent results on the secrecy capacity of the Gaussian multiple-input multiple-output channels in the presence of a multiantenna eavesdropper (MIMOME) in [4–6]. The first outer bound is a genie-aided bound that allows the source and relay to cooperate perfectly resulting in a Gaussian MIMOME channel for which jointly Gaussian inputs maximize the capacity. We show that these bounds are tight for a subclass of channels in which the multiaccess channel from the source and relay to the destination is the bottleneck link, and the eavesdropper is limited to receiving signals on the channel from the source and the relay to the destination. For a complementary subclass of channels in which the source-relay link is unusable due to noise resulting in adeaf relay, we develop a genie-aided bound where the relay and destination act like a two-antenna receiver. We also show that noise forwarding achieves this bound for this subclass of channels.
In [15], He and Yener study the secrecy rate of the channel studied here under the assumption that the relay is colocated with the eavesdropper, and the eavesdropper is completely cognizant of the transmit and receive signals at the relay. The authors found that using the relay does not increase the secrecy capacity, and hence there is no security advantage to using the relay. In this paper, we consider the eavesdropper as a separate entity and show that using the relay increases the secrecy capacity in some cases. In the model of [15], the eavesdropper can overhear only on the channel to the relay, while we consider three cases in which the eavesdropper can overhear on either or both the channels.
The paper is organized as follows. In Section 2, we present the channel models. In Section 3, we develop the inner and outer bounds on the secrecy capacity of the discrete
memoryless model. We illustrate these results with examples inSection 4. InSection 5, we present inner and outer bounds for the Gaussian channel model and illustrate our results with examples. We conclude inSection 6.
2. Channel Models and Preliminaries
2.1. Discrete Memoryless Model. A discrete-memoryless relay eavesdropper channel is denoted by (X1×X2,p(y,y1,y2 | x1,x2),Y×Y1×Y2) such that the inputs to the channel
in a given channel use areX1 ∈ X1 andX2 ∈ X2 at the
source and relay, respectively, the outputs of the channel are Y1 ∈ Y1,Y ∈ Y, and Y2 ∈ Y2, at the relay, destination,
and eavesdropper, respectively, and the channel transition probability is given by pY Y1Y2|XX2(y,y1,y2 | x,x2) [11]. The
channel is assumed to be memoryless; that is, the channel outputs at timeidepend only on channel inputs at time i. The source transmits a messageW1∈W1= {1, 2,. . .,M}to
the destination using the (M,n) code consisting of
(1) a stochastic encoder f at the source such that f : W1 → X1n∈Xn1,
(2) a set of relay encoding functions fr,i : (Y1,1,Y1,2,. . ., Y1,i−1)→ x2,iat every time instanti, and
(3) a decoding function at the destinationΦ:Yn →W
1.
The average error probability of the code is defined as
Pn e =
w1∈W1
1
MPr{Φ(Y
n)
/
=w1|w1was sent}. (1)
The equivocation rate at the eavesdropper is defined as Re = (1/n)H(W1 | Y2n). A perfect secrecy rate of R1 is
achieved if, for any > 0, there exists a sequence of codes (M,n) and an integerNsuch that, for alln≥N, we have
R1=
1 nlog2M, Pen≤, 1
nH(W1|Y2)≥R1−.
(2)
The secrecy capacity is the maximum rate satisfying (2). The model described above considers a relay that transmits and receives simultaneously in the same orthogonal channel. Inner and outer bounds for this model are developed in [11, Theorem 1].
In this paper, we consider a relay eavesdropper channel with orthogonal components in which the relay receives and transmits on two orthogonal channels. The source transmits on both channels, one of which is received at the relay and the other at the destination. The relay transmits along with the source on the channel received at the destination. Thus, the source signalX1consists of two partsXR∈XRandXD ∈ XD, transmitted to the relay and the destination, respectively, such thatX1 = XD ×XR. The eavesdropper can receive
Relay
Figure1: The relay-eavesdropper channel with orthogonal compo-nents.
orthogonal channeli,i=1, 2, andY2 =Y2,1×Y2,2. More
formally, the relay eavesdropper channel with orthogonal components is defined as follows.
Definition 1. A discrete-memoryless relay eavesdropper channel is said to have orthogonal components if the sender alphabetX1 =XD×XRand the channel can be expressed
as
Definition 1assumes that the eavesdropper can receive signals in both channels. In general, the secrecy capacity bounds for this channel depend on the receiver capabilities of the eavesdropper. To this end, we explicitly include the following two definitions for the cases in which the eavesdropper can receive signals in only one of the channels.
Definition 2. The eavesdropper is limited to receiving signals on the channel from the source to the relay, if
py,y1,y2,1,y2,2|xR,xD,x2
Definition 3. The eavesdropper is limited to receiving signals on the channel from the source and the relay to the destination, if
Remark 4. In the absence of an eavesdropper, that is, for y2,1 = y2,2 = 0, the channels in (3)–(5) simplify to that of
a relay channel with orthogonal components.
Thus, depending on the receiver capabilities at the eavesdropper, there are three cases that arise in developing the secrecy capacity bounds. For brevity, we henceforth identify the three cases as Cases1,2, and3, where Cases1and 2correspond to Definitions2and3, respectively, and Case 3is the general case where the eavesdropper receives signals from both the channels.
2.2. Gaussian Model. For a Gaussian relay eavesdropper channel with orthogonal components, the signalsY1andY
received at the relay and the destination, respectively, in each time symboli∈ {1,. . .,n}, are
Y1[i]=hs,rXR[i] +Z1[i], (6) Y[i]=hs,dXD[i] +hr,dX2[i] +Z[i], (7)
wherehk,mis the channel gain from transmitterk∈ {s,r}to receiverm∈ {r,d}, and whereZ1andZare zero mean unit
variance Gaussian random variables. The transmitted signals XR,XD, andX2are subject to average power constraints given
by
whereE[·] denotes expectation of its argument. The signals at the eavesdropper are
Y2,1[i]=hs,e,1XR[i]1e,1+Z2,1[i],
Y2,2[i]=hs,e,2xD[i]1e,2+hr,eX2[i]1e,2+Z2,2[i],
(9)
wherehs,e,1andhs,e,2are the channel gains from the source
to the eavesdropper in the two orthogonal channels,hr,e is the channel gain from the relay to the eavesdropper,Z2,1and Z2,2are zero-mean unit variance Gaussian random variables
assumed to be independent of the source and relay signals, and
1, if the eavesdropper can eavesdrop in orthogonal channel j=1, 2, 0, otherwise.
(10)
Throughout the sequel, we assume that the channel gains are fixed and known at all nodes.
For a relay channel with orthogonal components, El Gamal and Zahedi of show that a strategy where the source uses each channel to send an independent message and the relay decodes the message transmitted in its channel achieves capacity [14]. Due to the fact that the relay has partial access to the source transmissions, this strategy is sometimes also referred to as partial decode and forward (see [16]). The achievable scheme involves block Markov superposition encoding while the converse is developed using the max-flow, min-cut bounds. The following proposition summarizes this result.
Proposition 5([14]). The capacity of a relay channel with orthogonal component is given by
C=max min{I(XR;Y1|X2) +I(XD;Y|X2),
I(XRXDX2;Y)},
(11)
where the maximum is over all input distributions of the form
For the Gaussian model, the bounds in(11)are maximized by jointly Gaussian inputs transmitting at the maximum power and subject to(12).
Remark 6. While the converse allows for all possible joint distributions ofXR,XD, andX2, from the form of the mutual
information expressions in (11), it suffices to consider distributions only of the form given by (12).
We use the standard notation for entropy and mutual information [17] and take all logarithms to the base 2 so that our rate units are bits. For ease of exposition, we writeC(x) to denote (1/2) log(1 +x) and writex+to denote max(x, 0).
We also write random variables with uppercase letters (e.g., Wk) and their realizations with the corresponding lowercase letters (e.g.,wk). We drop subscripts on probability distributions if the arguments are lowercase versions of the corresponding random variables. Finally, for brevity, we henceforth refer to the channel studied here as the orthogonal relay eavesdropper channel.
3. Discrete Memoryless Channel:
Outer and Inner Bounds
In this section, we develop outer and inner bounds for the secrecy capacity of the discrete-memoryless orthogonal relay eavesdropper channel. The proof of the outer bounds follows along the same lines as that in [11, Theorem 1] for the full-duplex relay-eavesdropper channel and is specialized for the orthogonal model considered here. The following theorem summarizes the bounds for the three cases in which the eavesdropper can receive in either one or both orthogonal channels.
Theorem 7. An outer bound on the secrecy capacity of the relay eavesdropper channel with orthogonal components is given by the following.
Case 1.
Cs≤max[min{I(VDVR;Y Y1|V2U),I(VDV2;Y|U)}
−I(VR;Y2|U)]+.
(13)
Case 2.
Cs≤max[min{I(VDVR;Y Y1|V2U),I(VDV2;Y|U)}
−I(VDV2;Y2|U)]+.
(14)
Case 3.
Cs≤max[min{I(VDVR;Y Y1|V2U),I(VDV2;Y|U)}
−I(VRVDV2;Y2|U)]+,
(15) whereU,VD,VR, andV2are auxiliary random variables, and
the maximum is over all joint distributions satisfyingU →
(VR,VD,V2)→ (XR,XD,X2)→ (Y,Y1,Y2).
Proof. The proof is extended from the outer bound in [11, Theorem 1] to include auxiliary random variables corresponding to each of the transmitted signals and is developed inAppendix A.
Following Proposition 5, a natural question for the relay-eavesdropper channel with orthogonal components is whether the PDF strategy can achieve the secrecy capacity. To this end, we first develop the achievable PDF secrecy rates for the class offull-duplexrelay-eavesdropper channels and then specialize the result for the orthogonal model. The following theorem summarizes the inner bounds on the secrecy capacity achieved by PDF for the full-duplex (nonorthogonal) relay-eavesdropper channels.
Theorem 8. An inner bound on the secrecy capacity of a
full-duplexrelay eavesdropper channel, achieved using partial decode and forward, is given by
Cs≥min{I(X1;Y|X2,V) +I(V;Y1|X2),I(X1X2V;Y)}
−I(X1X2;Y2)
(16)
for all joint distributions of the form
p(v)p(x1|v)p(x2|v)p
y1,y|x1,x2
. (17)
Proof. The proof is developed inAppendix Band uses block Markov superposition encoding at the source such that, in each block, the relay decodes a part of the source message while the eavesdropper has access to both source messages.
The following theorem specializes Theorem 8 for the orthogonal relay-eavesdropper channel.
Theorem 9. An inner bound on the secrecy capacity of the orthogonal relay eavesdropper channel, achieved using partial decode and forward over all joint distributions of the form
p(xR,xD,x2), is given by the following.
Case 1.
Cs≥min{I(XDXR;Y Y1|X2),I(XDX2;Y)}
−I(XR;Y2).
(18)
Case 2.
Cs≥min{I(XDXR;Y Y1|X2),I(XDX2;Y)}
−I(XD,X2;Y2).
(19)
Case 3.
Cs≥min{I(XDXR;Y Y1|X2),I(XDX2;Y)}
−I(XR;Y2|X2)−I(XD,X2;Y2).
Proof. The proof is developed in Appendix Cand involves specializing the bounds in Theorem 8 for the orthogonal model. It is further shown that the input distribution can be generalized to all joint probability distributions p(xR,xD,x2).
The bounds in (20) can be generalized by randomizing the channel inputs. We now prove that PDF with random-ization achieves the secrecy capacity.
Theorem 10. The secrecy capacity of the relay channel with orthogonal complements is the following.
Case 1.
Cs=max[min{I(VDVR;Y Y1|V2U),I(VDV2;Y|U)}
−I(VR;Y2|U)]+.
(21)
Case 2.
Cs=max[min{I(VDVR;Y Y1|V2U),I(VDV2;Y|U)}
−I(VDV2;Y2|U)]+.
(22)
Case 3.
Cs≤max[min{I(VDVR;Y Y1|V2U),I(VDV2;Y|U)}
−I(VRVDV2;Y2|U)]+,
(23) whereU,VD,VR, andV2are auxiliary random variables, and
the maximum is over all joint distributions satisfyingU →
(VR,VD,V2) → (XR,XD,X2) → (Y,Y1,Y2). Furthermore,
for Case3,
Cs≥[min{I(VDVR;Y Y1|V2U),I(VDV2;Y |U)}
−I(VR;Y2|V2U)−I(VD,V2;Y2|U)]+
(24)
for all joint distributions satisfyingU → (VR,VD,V2) →
(XR,XD,X2) →(Y,Y1,Y2).
Proof. The upper bounds follow from Theorem 7. For the lower bound, we prefix a memoryless channel with inputs VR,VD, and V2 and transition probability p(xR,xD,x2 | vR,vD,v2) (this prefix can potentially increase the achievable
secrecy rates as in [2,11]). The time-sharing random variable Uensures that the set of achievable rates is convex.
Remark 11. In contrast to the nonsecrecy case, where the orthogonal channel model simplifies the cut-set bounds to match the inner PDF bounds, for the orthogonal relay-eavesdropper model in which the relay-eavesdropper receives in both channels, that is, when the orthogonal receiver restrictions at the relay and intended destination do not apply to the eavesdropper, in general, the outer bound can be strictly larger than the inner PDF bound.
In the following section, we illustrate these results with three examples.
4. Examples
Example 12. Consider an orthogonal relay eavesdropper channel withXR =XD =X2 = {0, 1}. The outputs at the
relay and destination are given by
Y1=XR, Y =XDX2, (25)
while the output at the eavesdropper is
Y2,1=XR (channel 1),
Y2,2=
⎧ ⎨ ⎩
1, ifXD≤X2,
0, otherwise, (channel 2).
(26)
Since the destination can receive at most 1 bit in every use of the channel, the secrecy capacity of this channel is at most 1 bit per channel use. We now show that this secrecy capacity can be achieved. In each channel use, let the source send bit w ∈ {0, 1}such thatXR = 0, XD = w, andX2 = 1.
SinceX2 =1, the receiver obtainswwhile the eavesdropper
receivesY2,1=0 andY2,2=1 irrespective of the value of bit w. Hence, a perfect secrecy capacity of 1 can be achieved.
The code design inExample 12did not require random-ization. We now present an example where randomization is necessary.
Example 13. Consider an orthogonal relay eavesdropper channel where all the input and output alphabets are the same and given by{0, 1}2. We writeX
R = (aR,bR), XD = (aD,bD), and X2 = (a1,b1) to denote the vector binary
signals at the source and the relay. The outputs of this channel, shown inFigure 2(a), at the relay, destination, and the eavesdropper are given by
Y=(aD,bD⊕a1), Y1=(aR,bR),
Y2,1=(aR,bR),
Y2,2=(a1,b1⊕aD),
(27)
where ⊕denotes the binary XOR operation. The capacity of this channel is at most 2 bits per channel use as the destination, via Y, can receive at most 2 bits per channel use. We will now show that a secrecy capacity of 2 bits per channel use can be achieved. Consider the following coding scheme. In every channel use, the relay flips an unbiased coin to generate a bitn∈ {0, 1}such that its transmitted signal is
X2=(0,n). (28)
In every use of the channel, the source transmits 2 bits, denoted asw1andw2, using
XR=(0, 0),
XD=(w1,w2).
Y1 aX2 1
b1
XR
aR
bR
aD
bD
XD
Y
Y2,2
Y2,1
(a)Example 13
Y1 aX2 1
b1
XR
aR
bR
aD
XD
Y
Y2,2
Y2,1
(b)Example 14
Figure2: Orthogonal relay eavesdropper channel model of Exam-ples13and14.
For these transmitted signals, the receiver and eavesdropper receive
Y=(w1,w2), Y2,1=(0, 0), Y2,2=(0,n⊕w1).
(30)
Thus, the receiver receives both bits while the eavesdropper is unable to decode any information due to the randomness ofn. This is an example where transmitting a random code from the relay is required to achieve the secrecy capacity.
In the above two examples, the source to relay link was completely available to the eavesdropper, and hence the relay could at best be just used to send random bits. In the next example, we show that the secrecy capacity is achieved by the relay transmitting a part of the message as well as a random signal.
Example 14. Consider an orthogonal relay eavesdropper channel where the input and output signals at the source, relay, and destination are binary two-tuples whileY2,1 and
Y2,2at the eavesdropper are binary alphabets. We writeXR= (aR,bR),XD = aD, and X2 = (a1,b1) to denote the vector
binary signals at the source and the relay. The outputs at the relay, destination, and the eavesdropper are also vector binary signals given by
Y=(a1,aD),
Y1=(aR,bR),
Y2,1=(bR),
Y2,2=(b1⊕aD),
(31)
as shown in Figure 2(b). As in the previous example, the capacity of this channel is also at most 2 bits per channel use. We now show that a secrecy capacity of 2 bits per channel use can be achieved for this example channel. Consider the following coding scheme: in theith use of the channel, the source encodes 2 bits, denoted asw1,iandw2,ias
XR=
w1,i, 0
, XD =
w2,i
. (32)
The relay receivesw1,i−1in the previous use of the channel.
Furthermore, in each channel use, it also generates a uniformly random bitni, and transmits
X2=
w1,i−1,ni
. (33)
With these transmitted signals, the received signals at the receiver and the eavesdropper are
Y=w1,i−1,w2,i
, Y2,1=(0),
Y2,2=
ni⊕w2,i
.
(34)
Thus, overn+ 1 uses of the channel the destination receives all 2n+ 1 bits transmitted by the source. On the other hand, in every use of the channel, the eavesdropper cannot decode either source bit.
5. Gaussian Model
0 1 2 3 4 5 6
Secr
ecy
rat
e
−1 −0.5 0 0.5 1 1.5 2 2.5
Location of the relay
Destination Eavesdropper Source
Upper bound—Case 1 Lower bound—Case 1 Direct link
(a) Case
0 1 2 3 4 5 6
Secr
ecy
rat
e
−1 −0.5 0 0.5 1 1.5 2 2.5
Location of the relay
Destination Eavesdropper Source
Upper bound—Band 2 Lower bound—Band 2 (PDF) Lower bound—Band 2 (NF) Wiretap channel
(b) Case
0 1 2 3 4 5 6
Secr
ecy
rat
e
−1 −0.5 0 0.5 1 1.5 2 2.5
Location of the relay
Destination Eavesdropper Source
Upper bound—Case 3 Lower bound—Case 3 (PDF) Lower bound—Case 3 (NF) Wiretap channel
(c) Case3
Figure3: Source is at (0, 0), destination is at (1, 0), and eavesdropper is at (1.5, 0). A distance fading model withα=2 is taken, and power constraints forXR,XD, andX2are all unity.
t×1 vector outputsYandYeat the intended destination and eavesdropper, respectively, given by
Y[i]=HX[i] +Z[i], (35) Ye[i]=HeX[i] +Ze[i], (36)
where in every channel usei,Z[i] andZe[i] are zero-mean Gaussian vectors with identity covariance matrices that are
independent across time symbols. The channel input satisfies an average transmit power constraint:
1 n
n
i=1
x2≤P. (37)
channels in (37), our outer bound requires a per antenna power constraint. To this end, we apply the results developed in [5] in which a more general transmitter covariance constraint is considered such that
1 n
n
i=1
x[i]xT[i]S, (38)
whereSis a positive semidefinite matrix, andABdenotes that B −A is a positive semidefinite matrix. The secrecy capacity of this channel is summarized in the following theorem.
Lemma 15 ([5]). The secrecy capacity of the MIMOME channel of (36)subject to(38)is given by
Cs= max
0KXS
1 2log det
I+HKXHT
−1
2log det
I+HeKXHTe
.
(39)
Remark 16. The expression in (39) can also be written as
Cs=max[I(X∗;Y)−I(X∗;Ye)], (40)
where he maximum is over allX∗∼N(0,KX).
We now present an outer bound on the Gaussian orthogonal relay eavesdropper channel usingLemma 15. Theorem 17. An outer bound on the secrecy capacity of the Gaussian orthogonal relay eavesdropper channel is given by the following.
Case 1.
Cs≤max[I(XDX2;Y)−I(XR;Y2)]. (41a)
Case 2.
Cs≤max[I(XDX2;Y)−I(XDX2;Y2)]. (41b)
Case 3.
Cs≤max[I(XDX2;Y)−I(XRXDX2;Y2)], (41c)
where the maximum is over all [XRXDX2]T ∼ N(0,KX)
whereKX=E[XXT] has diagonal entries that satisfy (8).
Remark 18. In (41a) and (41c), theXR∗maximizing the outer bound on the secrecy capacity isXR∗=0. On the other hand,
XR∗can be chosen to be arbitrary for (41b).
Proof. An outer bound on the secrecy capacity of the relay eavesdropper channel results from assuming that the source and relay can cooperate over a noiseless link without causality constraints. Under this assumption, the problem reduces to that of a MIMOME channel. Thus, applying
Lemma 15and using the form in (40), forX=[XRXDX2]T ∼
N(0,KX), the secrecy capacity can be upper bounded as
Cs≤max[I(XRXDX2;Y)−I(XRXDX2;Y2)] (42)
=max[I(XDX2;Y) +I(XR;Y|XDX2)−I(XRXDX2;Y2)]
(43)
=max[I(XDX2;Y)−I(XRXDX2;Y2)], (44)
where (44) follows from the orthogonal model in (3). Finally, applying the conditions on the eavesdropper receiver for the three cases simplifies the bounds in (44) to (41a), (41b), and (41c).
The PDF inner bounds developed in Section 3 for the discrete memoryless case can be applied to the Gaussian model with Gaussian inputs at the source and relay. In fact, for all three cases, the inner bounds require taking a minimum of two rates, one achieved jointly by the source and relay at the destination and the other achieved by the source at the relay and destination. Comparing the inner bounds in (19) with the outer bounds in (41b), for those channels in which the source and relay are clustered close enough that the bottle-neck link is the combined source-relay link to the destination and the eavesdropper overhears only the channel from the source and the relay to the destination, the secrecy capacity can be achieved. This is summarized in the following theorem.
Theorem 19. For a class of clustered orthogonal Gaussian relay channels with
I(XDX2;Y)< max
p(xR|xD,x2)
I(XDXR;Y Y1|X2), (45)
the secrecy capacity for Case2is achieved by PDF and is given by the following.
Case 2.
Cs=max[I(XDX2;Y)−I(XD,X2;Y2)], (46)
where the maximum is overX=[XR XD X2]T∼N(0,KX).
For a relay channel without secrecy constraints, the cut-set outer bounds are equivalent to two multiple-input multiple-output (MIMO) bounds, one that results from assuming a noiseless source-relay link and the other that results from assuming a noiseless relay-destination link. Under a secrecy constraint, the outer bound inTheorem 17 is based on the assumption of a noiseless source-relay link. The corresponding bound with a noiseless relay-destination link remains unknown.
capacity for this subclass and show that the noise-forwarding strategy introduced in [11] achieves this outer bound. Central to our proof is an additional constraint introduced in developing the outer bounds on the eavesdropper that does not decode the relay transmissions. Clearly, limiting the eavesdropper capabilities can only improve the secrecy rates, and thus, an outer bound for this channel with a constrained eavesdropper is also an outer bound for the original channel (with hs,r = 0 in both cases) with an unconstrained eavesdropper. We show that the outer bound for the constrained channel can be achieved by the strat-egy of noise-forwarding developed for the unconstrained channel.
Theorem 20. The secrecy capacity of a subclass of Gaussian orthogonal relay eavesdropper channels withhs,r =0for Cases 2and3is given by a communication link between the source and the relay, that is, hs,r = 0, the relay is oblivious to the source transmissions. Since the relay and the source do not share common randomness, one can setXR =0. Finally, sinceX2
depends onXDonly viaXRandXR=0,X2is independent of XD.
We now provide two outer bounds for fixedE[X2
D] and
E[X22]. Thus, the minimum of the two is also an outer bound.
Maximizing overE[XD2] andE[X22] will then yield the outer
bound as in the statement ofTheorem 20.
An outer bound on the secrecy capacity is obtained by applyingTheorem 17for Cases2and3as
Cs≤C
where (48) holds because XD and X2 are independent,
and the Gaussian signaling is optimal, which follows from Theorem 17.
We develop a second outer bound under the assumption that the relay and the destination have a noiseless channel such that they act like a two-antenna receiver. One can alternately view this as an improved channel that results
from having a genie that shares perfectly the transmitted and received signals at the relay with the destination. Since X2
is independent of XD, the destination can perfectly cancel
X2from its received signal, and thus, from (7), the effective
received signal at the destination can be written as
Y=hs,dXD+Z. (49)
On the other hand for the constrained eavesdropper, since the relay’s signalX2acts as interference and is independent
of XD, the information received at the eavesdropper is minimized whenX2is the worst case noise, that is, when it
is Gaussian distributed [18, Theorem II.1]. The equivalent signal received at the eavesdropper is then
Y2,2 =hs,e,2XD+
where Z2,2 is Gaussian with zero mean and unit variance.
Thus, the constrained eavesdropper channel simplifies to a MIMOME channel with a single-antenna source trans-mitting XD and single-antenna receiver and eavesdropper receiving Y and Y2,2 , respectively. For this channel, from
Lemma 15, the secrecy capacity of this constrained eaves-dropper channel is upper bounded as
Cs≤C
Finally, since (51) is an upper bound for the channel with an eavesdropper constrained to ignoreX2, it is also an upper
bound for the channel in which the eavesdropper is not constrained.
Inner Bound. The lower bound follows from the noise forwarding strategy introduced in [11, Theorem 3]. In this strategy, the relay sends codewords independent of the source message, which helps in confusing the eavesdropper. The noise forwarding strategy transforms the relay-eavesdropper channel into a compound multiple access channel, where the source/relay to the receiver is the first multiple access channel, and the source/relay to the eavesdropper is the second one.
5.2. Illustration of Results. We illustrate our results for the Gaussian model for a class of linear networks in which the source is placed at the origin, and the destination is unit distance from the source at (1, 0). The eavesdropper is at (1.5, 0). The channel gainhm,k, between transmitterm and receiverk, for each mandk, is modeled as a distance dependent path-loss gain given by
hm,k= 1 dα/m,2k
∀m∈ {s,r}, k∈ {r,d,e}, (52)
along the line connecting the source and the eavesdropper as shown inFigure 3. Furthermore, as a baseline assuming that the relay does not transmit, that is,XR =0, the secrecy capacity of the resulting direct link and the wire-tap channel for Cases2and3, respectively, are included in all three plots inFigure 3. The rates are plotted in separate subfigures for the three cases in which the eavesdropper receives signals in only one or both channels. In all cases, the path loss exponent αis set to 2, and the average power constraint onXR,XD, and
X2is set to unity. In addition to PDF, the secrecy rate achieved
by noise forwarding (NF) is also plotted.
InFigure 3, for all three cases, the PDF secrecy rates are obtained by choosing the input signal X = [XRXDX2]T to
be Gaussian distributed and numerically optimizing the rates over the covariance matrixKX=E[XXT] (more precisely the
three variances ofXR,XD,X2 and the pairwise correlation
among these three variables). We observe that the numerical results match the theoretical capacity result for Case2that PDF is optimal when the relay is close to the source. Further, the upper bounds for Cases2and3are the same as seen also in (41b)-(41c). On the other hand, when the relay is farther away than the eavesdropper and destination are from the source, there are no gains achieved by using the relay relative to the nonrelay wiretap secrecy capacity. Finally, for Cases2 and3, NF performs better than PDF when the relay is closer to the destination.
6. Conclusions
We have developed bounds on the secrecy capacity of relay eavesdropper channels with orthogonal components in the presence of an additional passive eavesdropper for both the discrete memoryless and Gaussian channel models. Our results depend on the capability of the eavesdropper to overhear either or both of the two orthogonal channels that the source uses for its transmissions. For the discrete memoryless model, when the eavesdropper is restricted to receiving in only one of the two channels, we have shown that the secrecy capacity is achieved by a partial decode-and-forward strategy.
For the Gaussian model, we have developed a new outer bound using recent results on the secrecy capacity of Gaussian MIMOME channels. When the eavesdropper is restricted to overhearing on the channel from the source and the relay to the destination, our bound is tight for a subclass of channels where the source and the relay are clustered such that the combined link from the source and the relay to the destination is the bottleneck. Furthermore, for a subclass where the source-relay link is not used, we have developed a new MIMOME-based outer bound that matches the secrecy rate achieved by the noise forwarding strategy.
A natural extension to this model is to study the sec-recy capacity of orthogonal relay channels with multiple relays and multiple eavesdroppers (see, e.g., [19]). Also, the problem of developing an additional outer bound that considers a noiseless relay destination link remains open for the channel studied here.
Appendices
A. Proof of
Theorem 7
In this section, we will prove the upper bounds on the secrecy capacity for all the three cases. Following a proof similar to that in [11, Theorem 1], we bound the equivocation as
nRe≤
Now, letJbe a random variable uniformly distributed over
{1, 2,. . .,n}and setU =JYi−1Yn
2,i+1,VR=JY2,i+1W1,VD =
JY2,ni+2W1,V2=JYi−1,Y1=Y1,J,Y2=Y2,J, andY=YJ. We specialize the bounds in (A.1) separately for each case.
Case 1. From (A.1), we have
Case 2. From (A.1), we have
This proves the upper bound for Case2.
Case 3. From (A.1), we have This proves the upper bound for Case3. For perfect secrecy, setting R1 = Re yields the upper bound on the secrecy capacity.
B. Proof of
Theorem 8: PDF for Relay
Eavesdropper Channel
Random Partition. Randomly partition the set {1, 2,. . ., 2nR1}into 2n(I(X2;Y)−)cellsS
m.
Encoding. Letwibe the message to be sent in blockiwhere the total number of messages is 2n(R1+R2−I(X1X2;Y2)). Further,
Decoding. At the end of blocki, we have the following. (1) The receiver estimates mi by looking at jointly
(2) The receiver calculates a setL1(y(i−1)) ofwsuch
thatw ∈L1(y(i−1)) if (v(w |mi−1),y(i−1)) are
jointly-typical. The receiver then declares thatwi−1
was sent in blocki−1 ifwi−1 ∈Smi∩L1(y(i−1)).
The probability thatwi−1=wi−1with arbitrarily high
probability provided that nis sufficiently large and R1< I(X2;Y) +I(V;Y |X2)−.
(3) The receiver declares thatwi−1was sent in blocki−1 if
(x1(wi−1 |mi−1,wi−1),y(i−1)) are jointly-typical.
wi−1 =wi−1 with high probability ifR2 =I(X1;Y | X2,V)−andnis sufficiently large.
(4) The relay upon receiving y1(i) declares thatwwas
received if (v(w | mi),y1(i),x2(mi)) are jointly -typical. wi = w with high probability if R1 < I(V;Y1 | X2) andn is sufficiently large. Thus, the
relay knows thatwi∈Smi+1.
Thus, we obtain
R1< I(X2;Y) +I(V;Y|X2)−, R1< I(V;Y1|X2),
R2=I(X1;Y|X2,V)−.
(B.8)
Therefore, the rate of transmission fromX1toY is bounded
by
R=R1+R2−I(X1X2;Y2)
=min{I(X1;Y |X2,V) +I(V;Y1|X2),I(X1X2V;Y)}
−I(X1X2;Y2).
(B.9)
Equivocation Computation. From [11, Theorem 2, equation (41)], we have
H(W1|Y2)≥H(X1)−I(X1,X2;Y2)−H(X1,X2|W1,Y2).
(B.10) ConsiderH(X1,X2 | W1,Y2). Since we knowW1, the only
uncertainty is the knowledge of li which can be decoded fromY2with arbitrarily small probability of error sinceli ∈
{1,. . ., 2nI(X1X2;Y2)}. Hence,
H(W1|Y2)≥n(R1+R2)−I(X1,X2;Y2)=nR (B.11)
thus givingRe=R, and hence we get perfect secrecy. Thus, the secrecy rate is given by
R=min{I(X1;Y X2,V) +I(V;Y1X2), I(X1X2V;Y)} −I(X1X2;Y2).
(B.12)
C. Proof of
Theorem 9: PDF for Relay
Eavesdropper Channel with
Orthogonal Components
FromTheorem 8, a secrecy rate of
R=min{I(X1;Y|X2,V) +I(V;Y1|X2), I(X1X2V;Y)} −I(X1X2;Y2)
(C.13)
can be achieved by partial decode and forward. LetX1 =
(XR,XD) andV = XR such that the input distribution is of the form p(x2)p(xR |x2)p(xD | x2). The achievable secrecy
rate is then given by
R=min{I(XRXD;Y |X2,XR) +I(XR;Y1|X2), I(XRXDX2;Y)} −I(XRXDX2;Y2)
(C.14)
=min{I(XD;Y|X2,XR) +I(XR;Y1|X2), I(XDX2;Y)} −I(XRXDX2;Y2)
(C.15)
=min{I(XD;Y|X2) +I(XR;Y1|X2), I(XDX2;Y)} −I(XRXDX2;Y2).
(C.16)
The equality in (C.16) follows from the fact thatXD−X2− XR is a Markov chain. We further specialize the bounds for the three cases based on the receiving capability of the eavesdropper.
Case 1.
R=min{I(XD;Y |X2) +I(XR;Y1|X2), I(XDX2;Y)} −I(XR;Y2).
(C.17)
The maximization of the expression to the right of the equality in (C.17) overp(xD,xR,x2)=p(x2)p(xR|x2)p(xD|
x2) is equivalent to maximizing over the more general
distribution p(xD,xR,x2), and henceforth, without loss of
generality we consider the general probability distribution p(xD,xR,x2).
We now prove that I(XD;Y | X2) +I(XR;Y1 | X2) ≥ I(XDXR;Y Y1 | X2) which completes the proof of this part
of the theorem. We have I(XD;Y|X2) +I(XR;Y1|X2)
=H(Y|X2)−H(Y |X2XD) +I(XR;Y1|X2)
≥H(Y|X2Y1)−H(Y |X2XD) +I(XR;Y1|X2)
=H(Y|X2Y1)−H(Y |X2XDXRY1) +I(XR;Y1|X2)
=I(Y;XDXR|X2Y1) +I(XR;Y1|X2)
=I(Y;XDXR|X2Y1) +I(XD;Y1|X2XR) +I(XR;Y1|X2)
=I(Y;XDXR|X2Y1) +I(XRXD;Y1|X2)
=I(Y Y1;XDXR|X2).
(C.18)
Case 2.
R=min{I(XD;Y |X2) +I(XR;Y1|X2), I(XDX2;Y)} −I(XDX2;Y2).
(C.19)
Note that maximization of above term over p(xD,
maximizing over generalp(xD,xR,x2), and henceforth,
with-out loss of generality we consider the general probability distributionp(xD,xR,x2).
We now prove thatI(XD;Y | X2) +I(XR;Y1 | X2) ≥ I(XDXR;Y Y1 | X2) which completes the proof of this part
of the theorem. We have
I(XD;Y |X2) +I(XR;Y1|X2)
=H(Y|X2)−H(Y |X2XD) +I(XR;Y1|X2)
≥H(Y|X2Y1)−H(Y|X2XD) +I(XR;Y1|X2)
=I(Y Y1;XDXR|X2),
(C.20)
where the last step follows as was shown earlier in (C.18).
Case 3.
R=min{I(XD;Y|X2) +I(XR;Y1|X2),I(XDX2;Y)}
−I(XR;Y2|X2XD)−I(XDX2;Y2)
=min{I(XD;Y|X2) +I(XR;Y1|X2),I(XDX2;Y)}
−I(XR;Y2|X2)−I(XDX2;Y2).
(C.21)
Note that maximization of above term over p(xD,
xR,x2) = p(x2)p(xR | x2)p(xD | x2) is equivalent to
max-imizing over generalp(xD,xR,x2).
We now prove thatI(XD;Y | X2) +I(XR;Y1 | X2) ≥ I(XDXR;Y Y1 | X2) which completes the proof of this part
of the theorem. We have
I(XD;Y |X2) +I(XR;Y1|X2)
=H(Y|X2)−H(Y |X2XD) +I(XR;Y1|X2)
≥H(Y|X2Y1)−H(Y|X2XD) +I(XR;Y1|X2)
=I(Y Y1;XDXR|X2),
(C.22)
where the last step follows as was shown earlier in (C.18).
Acknowledgments
The authors wish to thank Elza Erkip of Polytechnic Institute of NYU and Lifeng Lai and Ruoheng Liu of Princeton Uni-versity for useful discussions related to this paper. The work of V. Aggarwal and A. R. Calderbank was supported in part by NSF under Grant 0701226, by ONR under Grant N00173-06-1-G006, and by AFOSR under Grant FA9550-05-1-0443. The work of L. Sankar and H. V. Poor was supported in part by the National Science Foundation under Grant CNS-06-25637. The material in this paper was presented in part at the Information Theory and Applications Workshop, San Diego, CA, 2009 and at the IEEE International Symposium on Information Theory, Seoul, Korea, 2009.
References
[1] A. Wyner, “The wire-tap channel,” Bell System Technical Journal, vol. 54, no. 8, pp. 1355–1387, 1975.
[2] I. Csisz´ar and J. K¨orner, “Broadcast channels with confidential messages,”IEEE Transactions on Information Theory, vol. 24, no. 3, pp. 339–348, 1978.
[3] E. Tekin and A. Yener, “The general Gaussian multiple-access and two-way wiretap channels: achievable rates and coopera-tive jamming,”IEEE Transactions on Information Theory, vol. 54, no. 6, pp. 2735–2751, 2008.
[4] A. Khisti and G. W. Wornell, “The MIMOME channel,” in Proceedings of the 45th Annual Allerton Conference on Communication, Control, and Computing, Monticello, Ill, USA, September 2007.
[5] T. Liu and S. Shamai, “A note on the secrecy capacity of the multi-antenna wiretap channel,” IEEE Transactions on Information Theory, vol. 55, no. 6, pp. 2547–2553, 2009. [6] F. Oggier and B. Hassibi, “The secrecy capacity of the MIMO
wiretap channel,” in Proceedings of the IEEE International Symposium on Information Theory, pp. 524–528, Toronto, Canada, July 2008.
[7] M. Bloch and A. Thangaraj, “Confidential messages to a cooperative relay,” in Proceedings of the IEEE Information Theory Workshop (ITW ’08), pp. 154–158, Porto, Portugal, May 2008.
[8] Y. Liang and H. V. Poor, “Multiple-access channels with confidential messages,” IEEE Transactions on Information Theory, vol. 54, no. 3, pp. 976–1002, 2008.
[9] Y. Liang, H. V. Poor, and S. Shamai, “Secure communication over fading channels,” IEEE Transactions on Information Theory, vol. 54, no. 6, pp. 2470–2492, 2008.
[10] Y. Liang, A. Somekh-Baruch, H. V. Poor, S. Shamai, and S. Verd ´u, “Capacity of cognitive interference channels with and without secrecy,” IEEE Transactions on Information Theory, vol. 55, no. 2, pp. 604–619, 2009.
[11] L. Lai and H. El Gamal, “The relay-eavesdropper channel: cooperation for secrecy,” IEEE Transactions on Information Theory, vol. 54, no. 9, pp. 4005–4019, 2008.
[12] M. Yuksel and E. Erkip, “The relay channel with a wire-tapper,” in Proceedings of the 41st Annual Conference on Information Sciences and Systems (CISS ’07), pp. 13–18, March 2007.
[13] Y. Oohama, “Capacity theorems for relay channels with confidential messages,” inProceedings of the IEEE International Symposium on Information Theory, Nice, France, June 2007. [14] A. El Gamal and S. Zahedi, “Capacity of a class of relay
channels with orthogonal components,”IEEE Transactions on Information Theory, vol. 51, no. 5, pp. 1815–1817, 2005. [15] X. He and A. Yener, “On the equivocation region of relay
channels with orthogonal components,” in Proceedings of the 41st Annual Asilomar Conference on Signals, Systems and Computers, pp. 883–887, Pacific Grove, Calif, USA, November 2007.
[16] G. Kramer, “Models and theory for relay channels with receive constraints,” inProceedings of the 42nd Annual Allerton Confer-ence on Communication, Control, and Computing, Monticello, Ill, USA, September 2004.
[18] S. N. Diggavi and T. M. Cover, “The worst additive noise under a covariance constraint,” IEEE Transactions on Information Theory, vol. 47, no. 7, pp. 3072–3081, 2001.
